**Quantum
thermodynamics and the Gibbs paradox**

**Introduction, papers,
conferences, scientific press.**

Introduction

** Quantum thermodynamics**

A fundamental question is: What remains
of thermodynamics if one goes to the extreme limit of small quantum systems
with a few degrees of freedom? If it does survive, are the many formulations
of the second law (entropy of a closed system cannot decrease, heat goes from
high to low temperatures, the optimal changes are adiabatic ones,..) still equivalent,
or is there a "universal" formulation?

On this subject our group found many fundamental results, partly discovering
and defining the subject of Quantum Thermodynamics itself.

** First and second law**

The first law can still be defined,
provided the work scource and the bath are macroscopic [L38], [P51], [C29], [C27].
This implies that quantum thermodynamics exists as a non-trivial subject.

It has been realized that there is
one formulation of the second law which holds in general: Thomson's formulation (making cycles
costs work) for a system starting in equilibrium. It applies to systems without
bath, or systems coupled to a bath. The statement about Thomson's formulation
was known before; we reproduced it to draw attention to it and eleborated on
it [P58], [P62].

** "Appalling" predictions**

Several other formulations of the second law appear to be violated. The Clausius
inequality dQ less or equal to T dS
was shown to be invalid at T=0. The physical reason is the formation of a cloud
of bath modes around the central particle. Such clouds are well known in the
Kondo problem and for polarons. The energy of that cloud must be attributed
to the bath, and some of it can be taken out, because changing the parameters of
the particle changes the shape of the cloud [L38], [P51].

A test for the violation of the Clausius inequality was proposed for nanoscale
electric circuits [P59] and in quantum optics [L42], [C27].The Thomson formulation can be violated
when the system is still coupled to a single heat bath, but starts out of equilibrium.
Setups for such cycles were derived analytically [P51].

The rate of energy dissipation can be negative, even when starting in equilibrium
[P51]. Classically this would be forbidden, because there it is, after dividing
by temperature, equal to the rate of entropy production. Positivity of the latter
is another formulation of the second law.

The Landauer bound for information
erasure, sometimes said to be another formulation of the second law, can be
violated in the quantum regime [P52].

This was by some considered to be "appalling", but it is now checked by other groups
and becoming accepted generally.

** Application to quantum optics**

Also in quantum optics the manipulation
of the surrounding cloud can lead to surprising effects [L42], such as bath
assisted work extraction and bath assisted cooling [L46].

** Maximally extractable work**

It has been long thought that the maximal amount of work that can be extracted from a system is
determined by thermodynamic arguments, and in particular the entropy. In finite quantum systems,
this puts one constraint, but a macroscopic work source leads to a time-dependent Hamiltonian.
The generated unitary dynamics implies that there are more constraints,
namely all eigenvalues of the density matrix are conserved. Thus, generally less work can be extracted.
The maximal amount of extractable work is a new quantity, called * ergotropy*, and it has a simple
explantion: in the optimal final state is the largest amount of particles is in the ground state,
the one-but-largest in the first-excited state, and so on [L45].

** Explanation of the Gibbs paradox within the framework of Quantum Thermodynamics**

In 1875 the founding father of statistical physics Josiah Willard Gibbs pointed at the following paradox:
Take two equal volumina of different gases
and mix them. Then the entropy increases by and amount k log 2 per particle.
But if the gases are equal, there is not such an increase. The paradox lies in the discontinuity:
there is an increase no matter how small the difference between the gases, but not when they are equal.
This raises questions such as: if the gases are composed of similar balls, red ones for the first gas, blue ones
for the second, then what should a color-blind
experimentator conclude? In other words: the mixing entropy is not an operational concept.

There has been a long effort to resolve the paradox, which shows a limit of phenemenological thermodynamics.
It was believed to be solved by the quantum * mixing entropy* argument, but that was shown to create a
new problem at almost the same spot: it is a neither operational.

Assuming that the translational degrees of freedom of both gases are in thermal
equilibrium at the same temperature,
we express the differences between the gases by their internal (spin) structure.
The latter involve a
few degrees of freedom. Therefore we approach the problem via
quantum thermodynamics, the theory of thermodynamics for
small quantum systems connected to a macroscopic bath and a macroscopic work source.
In this field we notioced before that the notion of entropy is messy,
* the physical quantity is work*.
The maximal amount of work that can generally be extracted from a finite quantum system was already
derived in a paper with R. Balian: the so-called * ergotropy*.
This allows to consider the maximal amount of work that can be derived before and after mixing.
The difference is the * mixing work* or * mixing ergotropy*. Like the
mixing entropy, the mixing work is continuous when the gasses become more and more equal.
But the extractable work is indeed an operational concept, it depends on the
work extraction process employed.

This explains the Gibbs paradox using quantum mechanics alone [P70].

Papers

[P72] Armen E. Allahverdyan and Theo M. Nieuwenhuizen,

**Minimal Work Principle and its Limits for Classical Systems,**

Phys. Rev. E **75**, 051124 (2007) (5 pages).

[P70] A.E. Allahverdyan and Th.M. N.,
**Explanation of the Gibbs paradox within the framework of quantum thermodynamics**,
Phys. Rev. E **73**, 066119,1-15 (2006) .

[L46] A.E. Allahverdyan, R. Serral Gracia and Th. M. N., **Bath Assisted Cooling of Spins**,
Phys. Rev. Lett. 93, 260404,1-4, (2004)

[L45] Armen E. Allahverdyan,
Roger Balian and Theo M. N., ** Maximal work extraction from finite quantum systems**,
Europhys. Lett., **66**(2004) 419-422

[P66] A.E. Allahverdyan and Th.M. N.,

[P62] A.E. Allahverdyan, R. Balian
and Th. M. N., **Thomson's formulation of the second law for macroscopic and
finite work sources ** Entropy 6, (2004) 30-37 Entropy 6, (2004) 30-37

[C29] A.E. Allahverdyan, R. Balian
and Th. M. N., **Quantum thermodynamics: thermodynamics at the nanoscale **
Proceedings of Physics of Quantum Electronics XXXIV (PQE 2004), Journal of Modern
Optics (2004); cond-mat/0402387

[C27] Th. M. N., **Thermodynamics
and small quantum systems, ** Proceedings of Physics of Quantum Electronics
XXXIII (PQE 2003), Journal of Modern Optics 50, 2433-2442 (2003); cond-mat/0311582

[C30] Th.M. N., Armen E.
Allahverdyan, and Roger Balian, ** Mesoscopic perpetuum mobile of the second
kind,** preprint ITFA-2002-30

[P60] A.E. Allahverdyan,
R. Balian, Th.M. N., ** Extracting work from a macroscopic thermal bath via
a mesoscopic work source**, preprint ITFA-2002-20

[L42] Armen E. Allahverdyan
and Th.M. N., ** Bath-generated work extraction and inversion-free gain in
two-level systems,** J. Phys. A: Math.
Gen. ** 36 **, (2003) 875-882

[P59] A.E. Allahverdyan
and Th.M. N., ** On testing the violation of the Clausius inequality in nanoscale
electric circuits**, Phys.
Rev. B ** 66 **, 115309 (2002) Also in: Virtual Journal of Nanoscale
Science & Technology, September 23, 2002, Volume 6, Issue 13

[P58] A.E. Allahverdyan
and Th.M. N., ** A mathematical theorem as the basis for the second law: Thomson's
formulation applied to equilibrium, ** Physica A 305, (2002) 542-552

[P52] A.E. Allahverdyan
and Th.M. N., **Breakdown of the Landauer bound for information erasure in
the quantum regime,** Phys. Rev. E ** 64,** 056117 (2001) (9 pages)

[P51] Th.M. N. and A.E.
Allahverdyan, ** Statistical thermodynamics of quantum Brownian motion: Construction
of perpetuum mobile of the second kind,** Phys. Rev. E ** 66, **036102 (2002) (52 pages)

[L38] A.E. Allahverdyan
and Th.M. N., ** Extracting work from a single thermal bath in the quantum
regime, ** Phys. Rev. Lett. ** 85** (2000) 1799-1802

**Conferences with a session on Quantum Thermodynamics**

**Conference FQMT'04 Prague 2004**

I was chairman of the
scientific committee of the conference: ** Frontiers of Quantum and Mesoscopic
Thermodynamics**, July 26-29 2004 in Prague. See the
webpage.

**Lorentz workshop Leiden, 2003**

I was co-organizer
of the workshop: ** Hot Topics in Quantum Statistical Physics: q-Thermodynamics,
q-Decoherence and q-motors**,
August 11-16, 2003, Leiden.

**Conference in San Diego, 2002**

I was coorganizer of:
the ** First International conference on Quantum limits to the second law**,
July 29-31 2002 at the University
of San Diego

[C26] Armen E. Allahverdyan,
Roger Balian and Theo .M. N., **Thomson's formulation of the second law: an
exact theorem and limits of its validity, ** in: * Quantum Limits to the
Second Law,* AIP Conf. Proc. Vol. 643 (2002), pp. 35-40,
cond-mat/0208563

[C25] Theo M. N. and Armen
E. Allahverdyan, ** Quantum Brownian motion and its conflict with the second
law, ** in: * Quantum Limits to the Second Law, * AIP Conf. Proc. Vol.
643 (2002), pp. 29-34,
cond-mat/0208564

[C24] Claudia Pombo, Armen
E. Allahverdyan, and Theo M. N., ** Bath generated work extraction in two-level
systems,** in: * Quantum Limits to the Second Law, * AIP Conf. Proc.
Vol. 643 (2002), pp. 254-258,
cond-mat/0208565

[C23] Theo M. N. and Armen
E. Allahverdyan, ** Unmasking Maxwell's Demon,** in: *Quantum Limits to
the Second Law, * AIP Conf. Proc. Vol. 643 (2002), pp. 436-441

**Scientific press**

A number of accounts have been devoted to our research:

About an improved setup for the
classical heat engine:

A new work ethic, Nature July 12, 2000;

New frontiers of thermodynamics, The American Institute of Physics July 17, 2000

About perpetuum mobile of the second
kind:

New frontiers of thermodynamics, The American Institute of Physics July 17,
2000

Over perpetuum mobile van de tweede
soort:

``Mazen in Gods grondwet: Een duivelse machine'',
De Volkskrant, 22 juli 2000

About perpetuum mobile:

``Breaking the law: Can quantum mechanics + thermodynamics = perpetual motion
?'',
Science News, October 7, pp 158, 2000

Sul moto perpetuo:

``Moto perpetuo'', Focus N. 100, Febbraio 2001, (Milano), pagina 36